Step 1: Understanding the Concept:
The time period of a mass-spring system is determined by the mass and the stiffness (spring constant) of the spring.
A crucial property of springs is that the spring constant \( k \) is inversely proportional to its natural length \( L \), expressed as \( k \propto \frac{1}{L} \).
When a spring is cut or combined in different configurations (series or parallel), the effective spring constant changes, which in turn alters the time period of oscillation.
Step 2: Key Formula or Approach:
1. Time period of a spring-mass system: \( T = 2\pi \sqrt{\frac{m}{k}} \).
2. Spring constant for a part of a spring: If a spring of constant \( k \) is cut into \( n \) equal parts, the constant of each part is \( k' = nk \).
3. Parallel combination of springs: The equivalent spring constant \( K_{eq} \) is the sum of individual constants: \( K_{eq} = k_1 + k_2 + ... + k_n \).
Step 3: Detailed Explanation:
Let the initial spring have a constant \( k \). The initial time period is:
\[ T = 2\pi \sqrt{\frac{m}{k}} \dots (i) \]
The spring is cut into three equal parts. The spring constant of each individual part becomes:
\[ k' = 3k \]
Now, these three identical parts are connected in parallel. The new equivalent spring constant \( K_{p} \) is:
\[ K_{p} = k' + k' + k' = 3k + 3k + 3k = 9k \]
The new time period \( T' \) for the same mass \( m \) is:
\[ T' = 2\pi \sqrt{\frac{m}{K_{p}}} = 2\pi \sqrt{\frac{m}{9k}} \]
We can simplify this as:
\[ T' = \frac{1}{\sqrt{9}} \left( 2\pi \sqrt{\frac{m}{k}} \right) = \frac{1}{3} T \]
Step 4: Final Answer:
The new time period of the mass will be \( T/3 \).